Volume et entropie minimale des espaces localement sym�triques

Entropy and systoles on surfaces

“…This amounts to defining the normalized volume entropy h vol (M, g) · Vol(M, g) 1 n , which is invariant under scaling. The infimum of the normalized volume entropy over the space of all metrics has been studied by A. Katok in [10], M. Gromov in [6], I. Babenko in [1] and G. Besson, G. Courtois and S. Gallot in [2]. These authors specifically found universal lower bounds (i.e., bounds which do not depend on the metric) on the normalized volume entropy.…”

Entropies, Volumes, and Einstein Metrics

“…In fact, a candidate for such an intermediate invariant, based on the minimal eigenvalue of the Laplacian on the universal covering, occurs in the work of Besson-Courtois-Gallot [8]. We now elaborate on this to discuss the following alternative to (1):…”

“…The inequalities involving T (M) are from [8], where T (M) was defined. We now explain the terms in (1), and the sources for the different inequalities.…”

Perelman’s $${\bar{\lambda}}$$ -invariant and collapsing via geometric characteristic splittings

“…The latter case was shown by Besson, Courtois and Gallot in a series of papers which found deep connections and consequences in the theory, the interested reader is invited to consult [3,4].…”

“…The minimal entropy h(M) is related to the minimal volume MinVol(M), volume entropy λ(M) and simplicial volume ||M|| of M in the following string of inequalities, noticed by Gromov, Manning and others [10,20,4,21] …”